An action of a discrete abelian group is called Anosov if it has at least one Anosov element. More generally, an abelian action A is Anosov if it has at least one partially hyperbolic element with the central foliation being the orbit foliation of A.
Higher rank Anosov actions turned out to be extremely rigid. Namely, in 1994 A.Katok and R.Spatzier proved that any smooth action whose generators are sufficiently close to those of a genuinely higher rank algebraic Anosov action A is smoothly conjugate to A.
In the first lecture we discuss examples of algebraic Anosov actions and outline the proof of local rigidity theorem. The main ingredients are foliation and cocycle rigidity results. The latter one is treated by case-by-case analysis and we will omit the proof of it. Proving foliation rigidity result is the subject of subsequent lectures.